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Properties

Label 16.4.76087965578...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{40}\cdot 5^{8}\cdot 11^{6}$
Root discriminant $31.09$
Ramified primes $2, 5, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20, 0, 56, 224, 420, -40, -64, -72, -794, 664, 432, -328, -128, 92, 4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 4*x^14 + 92*x^13 - 128*x^12 - 328*x^11 + 432*x^10 + 664*x^9 - 794*x^8 - 72*x^7 - 64*x^6 - 40*x^5 + 420*x^4 + 224*x^3 + 56*x^2 - 20)
 
gp: K = bnfinit(x^16 - 8*x^15 + 4*x^14 + 92*x^13 - 128*x^12 - 328*x^11 + 432*x^10 + 664*x^9 - 794*x^8 - 72*x^7 - 64*x^6 - 40*x^5 + 420*x^4 + 224*x^3 + 56*x^2 - 20, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 4 x^{14} + 92 x^{13} - 128 x^{12} - 328 x^{11} + 432 x^{10} + 664 x^{9} - 794 x^{8} - 72 x^{7} - 64 x^{6} - 40 x^{5} + 420 x^{4} + 224 x^{3} + 56 x^{2} - 20 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(760879655786905600000000=2^{40}\cdot 5^{8}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.09$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{50} a^{14} - \frac{1}{25} a^{12} + \frac{1}{50} a^{11} - \frac{4}{25} a^{10} + \frac{1}{25} a^{9} - \frac{2}{25} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{11}{25} a^{5} + \frac{2}{5} a^{4} + \frac{1}{25} a^{3} + \frac{8}{25} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{100083457103632250} a^{15} + \frac{295233051867533}{50041728551816125} a^{14} + \frac{3379865747039113}{100083457103632250} a^{13} - \frac{2838565505078671}{100083457103632250} a^{12} - \frac{97036943933116}{50041728551816125} a^{11} - \frac{20795361662161271}{100083457103632250} a^{10} + \frac{6580713272770203}{100083457103632250} a^{9} - \frac{20252509444985489}{100083457103632250} a^{8} - \frac{1649991790500938}{10008345710363225} a^{7} - \frac{5468920679635346}{50041728551816125} a^{6} - \frac{13969911690535011}{50041728551816125} a^{5} - \frac{16661999321193684}{50041728551816125} a^{4} - \frac{12053909231793031}{50041728551816125} a^{3} + \frac{10349687258229418}{50041728551816125} a^{2} - \frac{313194068689023}{10008345710363225} a - \frac{2305090250459677}{10008345710363225}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 281106.193207 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 4.2.1600.1, 4.2.17600.2, 8.4.4956160000.6

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$